I have a question concerning the Schrödinger model for the Weil representation over a finite field $\mathbb{F}_q$.

The way to present the action of the Weil representation $\omega$ of $Sp(2n,\mathbb{F}_q)$ in this model, is to give the action of matrices $\left(\begin{smallmatrix}a & 0\\ 0&^{t}a^{-1}\end{smallmatrix}\right)$ in a standard Levi, $\left(\begin{smallmatrix}1 & u\\ 0& 1\end{smallmatrix}\right)$ of a radical unipotent, and finally of matrices of the form $\left(\begin{smallmatrix}0 & -1\\ 1 & 0 \end{smallmatrix}\right)$. The question I have is about the Legendre symbol appearing in the formula for the action of elements of the Levi. The formula is
$$
\omega\left(\begin{matrix}a & 0\\ 0&^{t}a^{-1}\end{matrix}\right)f(x)=\left(\frac{det(a)}{q}\right)f(^tax)
$$
I do not understand where the Legendre symbol $\left(\frac{\cdot}{q}\right)$ comes from. In the p-adic setting this is replaced by $|det(a)|^{1/2}$, and the reason is so that the action of this Levi is unitary. I hope someone can clarify this doubt.

$\begingroup$What you mean by the Legendre symbol is the quadratic character on $\mathbf F_q^\times$, right? The notation you use for it looks like a Jacobi symbol mod $q$, which is not the same thing (e.g., different domains) if $q$ is not prime.$\endgroup$
– KConradMay 14 '18 at 19:49

1

$\begingroup$Hi KConrad, yes it is the quadratic character. I found that notation in a paper from R. Howe. I thought it was standard.$\endgroup$
– Jesua Israel Epequin ChavezMay 14 '18 at 20:08

1 Answer
1

As a way to "discover" the quadratic/norm characters appearing in the finite-field Segal-Shale-Weil/oscillator repn, as well as in that over local fields (with even-dimensional vector space so that the repn is a genuine linear repn rather than projective), one can imagine that a more naive guess for the group action by parabolic $P$ and Weyl element $w$ is correct, up to some correction factors. Then the correction factors can be found by re-expressing a product $g_1g_2$ with $g_1={p_1}wn_1$ and $g_2={p_2}wn_2$ of big-cell group elements as a big-cell element $g_1g_2=p_3wn_3$, and writing out the requirement that the repn be a group homomorphism.

The discrepancy is an exponential sum, which is relatively easily shown to be the character. The computations are a bit tedious, but straightforward.

The Stone-vonNeumann theorem shows that the actual range of possibilities is just that of a choice of constant, so any single computation determining the "adjustment" is sufficient.

$\begingroup$Anent the re-expression, MO discovered the related question "Open cell decomposition after applying a Weyl group element", on which a certain user described how to do exactly this: mathoverflow.net/a/152231 .$\endgroup$
– LSpiceJul 10 '18 at 19:31